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Educationcrystal systems: nomenclature and conventions

28th Jan 2018 13:40 UTCDaniel Atencio

https://www.ige.unicamp.br/terraedidatica/v13_3/PDF13_3/Td-13-3-212-9.pdf

28th Jan 2018 14:47 UTCJohan Kjellman Expert

From the abstract this looks like an interesting article for further discussion. Why not publish an English version? Or is this just a new version of Pecock 1950?

cheers

28th Jan 2018 14:50 UTCPaul Brandes 🌟 Manager

Agree.

Looks like a great article for someone who speaks Portuguese, not so much for an English speaker....

28th Jan 2018 14:54 UTCJohan Kjellman Expert

same conclusions/suggestions as in this one?

http://www.minsocam.org/ammin/AM35/AM35_882.pdf

29th Jan 2018 20:18 UTCDaniel Atencio

Dear All,

Many thanks for your interest!

Here is an English translation, probably full of errors:

Introduction

Crystallography is a science full of rules (international conventions) that have been made so that people from all over the world use the same crystallographic orientations and nomenclatures, so that they can understand each other. If each used different rules to orient the crystals and classify them, for example, as for the point group, crystal class and crystal system, Crystallography would become very complex. These issues are not generally very clear in the introductory books on Crystallography, such as those for application in Mineralogy such as Bloss (1971), Chvátal (2007) and Klein & Dutrow (2012). On the other hand, the current conventions, standardized by the IUCR (International Union of Crystallography), are sometimes inappropriate. For example, by naming triclinic a system in which not necessarily all angles of the unit cell are different from 90°, or by denominating monoclinic crystals that which may have angles all equal to 90 °.

Fundamentals

One thing that attracts a lot of attention to crystals is the presence of flat faces. These faces can be repeated a few times, rotating relative to each other (they are identical faces and can be seen after a rotation of the crystal), or they can be reflected (one is the mirror image of the other), or inverted (one is the reverse of the other). This repetition is known as symmetry. To describe the symmetry, some fictitious elements, called symmetry elements, can be used which allow us to imagine how the faces of the crystals "reproduce" themselves. One is the symmetry axis, an imaginary line, around which faces appear repeated 2, 3, 4 or 6 times. Another element of symmetry is the symmetry plane, which acts as a mirror and causes the faces to appear in pairs, one being the mirror image of the other. A third element is the symmetry center, which, when present, lies in the geometric center of the crystal and makes all its faces appear on the opposite side of the crystal. The last symmetry element is the improper symmetry axis, which with each rotation operation reverses the face of the crystal. These elements are represented in Figure 1.

Each crystal has an association of symmetry elements. For example, imagine a crystal with the shape of a perfect cube. In it we will find 3 symmetry axes of order 4, 4 improper symmetry axes of order 3, 6 symmetry axes of order 2, and 9 planes of symmetry. We also verified that this crystal has a symmetry center. This, then, is the association of symmetry elements (=symmetry degree) of the crystal whose shape is called a cube (Figure 2).

There are only 32 possibilities of associations of symmetry elements, also called point groups (group of symmetry elements around a point, located in the geometric center of the crystal). It is said that all crystals having the same point group belong to the same crystal class. And the crystal classes are grouped in crystal systems. For example, it is said that the crystal classes which have a symmetry axis of order 6 (proper or improper) belong to the hexagonal crystal system. All of this information is summarized in Table 1.

In Table 1, we can see something that seems strange: there is a monoclinic and a triclinic system, but there is no diclinic system... Let's talk about this soon.

In order to give "names" to the faces and directions of the crystals, they are oriented spatially. For this, another set of imaginary axes, the crystallographic axes, is used. These axes are in number of 3 or 4, depending on the crystal system. They intersect forming certain interaxial angles. The set between crystallographic axes and their respective interaxial angles is called the axial cross (Figure 3).

But why do crystals exhibit this external symmetry? Because of an internal symmetry! The atoms are arranged inside a crystal and if we observe their distribution in the same, we will see that they are distributed neatly, forming like "bricks" that construct a wall. These bricks, which repeatedly form a whole crystal structure, are called the unit cell. The dimensions of the unit cell (that is, the edges of the brick) are called unit cell parameters. Since the edges of the cell coincide with the crystallographic axes, the letters used to define the crystallographic axes, which are dimensionless, are the same as those applied to the dimensions of the unit cell. This may eventually cause some confusion.

The crystal systems are presented in two aspects in the introductory books of crystallography: a) with respect to the set of symmetry elements, and b) with respect to the axial cross.

Symmetry elements

As for the first aspect, the symmetry elements, we have the information of Table 2.

Axial cross

As for the second aspect, the axial cross, the information is usually presented as in table 3:

Application

Let us then try to apply the contents of Tables 1, 2 and 3 to some examples of minerals whose information regarding unit cell and crystal class parameters are given below in Tables 4, 5 and 6. We will exclude from our discussions the crystal compounds of trigonal and hexagonal systems to focus on "poorly standardized" crystal systems.

The minerals in Table 4 show the unit cell parameters a, b and c different from each other, two angles different of 90 ° and one equal to 90 °. They would belong, according to their axial cross, to the diclínico system, name that is not in the previous tables, but that already has been used in the crystallographic literature (see discussion in Roger, 1938, and Pertlik, 2006). In the monoclinic system, only one of the angles is different from 90o and in the triclinic, the three angles are different from 90o. When two angles are different from 90o, the system should be diclinic, but these minerals formally fit into the triclinic system, since they all belong to classes 1 and 2. What defines the crystal system are the crystal classes, not the axial cross.

The minerals in Table 5 show the unit cell parameters a, b and c different from each other, two angles equal to 90o and one different from 90o. In the monoclinic system, only one of the angles is different from 90o and in the triclinic, the three angles are different from 90o. They would belong, according to their axial cross, to the monoclinic system, but formally fit into the triclinic system, since all belong to classes 1 and -1

The minerals in Table 6 show the unit cell parameters a, b and c different from each other and the three angles equal to 90o. In the orthorhombic system all angles are equal to 90o and in the monoclinic system only one of the angles is different from 90o. They would belong, according to their axial cross, to the orthorhombic system, but formally fit into the monoclinic system, since all belong to classes 2, m and 2/m.

It is necessary to make a series of generalizations. For linear constants, if a (a1, a2, a3), b and c are treated as crystallographic axes: it is correct to say that the equals sign means "symmetrically equivalent" and the different sign means "symmetrically non-equivalent." If the same constants are thought of as unit cell dimensions: the meaning of the equals sign will be correct and that of different will mean "generally different", since the values can be the same, just by coincidence. For the angular constants, in both cases the meaning of the equals sign will be correct and the different one will mean "generally different".

1 - TRICLINIC SYSTEM: a, b and c are not equivalent to each other. Normally, the angles α, β and γ ≠ 90o, but may be equal to 90°, by coincidence. We see then that the term triclinic will no longer make sense! The dimensions of the unit cell parameters a, b and c will normally be different from each other, but coincidences may also occur. If one of the angles is equal to 90 ° and the dimensions a, b and c are different, or have some equivalence by coincidence, the crystal will be pseudo-diclinic * (Table 4). If two of the angles are equal to 90 ° and the dimensions a, b and c are different, or have some equivalence by coincidence, the crystal will be pseudo-monoclinic * (Table 5). If α = β = γ = 90 ° and the dimensions a, b and c are different, the crystal will be pseudo-ortorrômbic. If α = β = γ = 90º and two dimensions are equal, the crystal will be pseudo-tetragonal. And in a case where α = β = γ = 90° and the dimensions of a, b and c are equal, the crystal will be pseudo-cubic. In practice, the term "pseudo" is not only used when the angles are exactly 90 °, but also when they are very close to 90 °.

* Note that in cases of pseudo-diclinic and pseudo-monoclinic crystals it is the angle that necessarily needs to meet the specified requirements, and the parameters may or may not be different (only by coincidence). Although no mineralogical example to date presents such coincidences as to the equivalence between parameters, nothing prevents it from occurring.

2 – MONOCLINIC SYSTEM: a, b and c are not equivalent to each other. The angles α = γ = 90°. β ≠ 90o, but may be the same, coincidentally. The dimensions of the unit cell parameters a, b and c will normally be different from each other, but coincidences may also occur. If β = 90 ° and the dimensions a, b and c are different, the crystal will be pseudo-orthorhombic (Table 6). If, in addition to β = 90°, two dimensions of the cell are equal, the crystal will be pseudo-tetragonal. And in a hypothetical case where β = 90° and the dimensions of a, b and c are equal, the crystal will be pseudo-cubic.

3 - ORTHORHOMBIC SYSTEM: the axial cross is formed by 3 non-equivalent axes, a, b and c. The angles α = β = γ = 90°, necessarily. a, b and c, being non-equivalents, will usually exhibit different dimensions. However, coincidentally, two dimensions may be the same, leading to pseudo-tetragonal crystals. If the three dimensions coincidentally are the same, the crystals will be pseudo-cubic.

4 - TETRAGONAL SYSTEM: a and b are equivalent to each other and c is different (a1 = a2 ≠ c) and the angles α = β = γ = 90°, necessarily. Consequently, the dimensions of a and b (the parameters of the unit cell) must be equal. c will normally have a different dimension, but may coincidentally be of the same size as a and b, although not equivalent (do not coincide with the same symmetry elements as a and b). If the dimension of c is equal (or approximately equal) to that of a and b, the crystal will be pseudo-cubic.

5 - CUBIC SYSTEM: The crystallographic axes a, b and c must be equivalent to each other for the symmetry elements that coincide with them, The cell parameters (a1 = a2 = a3) and the angles α = β = γ = 90 °, also (obligatorily).

Therefore, what is fundamental for the definition of the crystal system is the set of symmetry elements and not the parameters of the unit cell or the angular relation. These are then some of the inappropriate conventions. It seems logical to think that triclinic crystals are those that have all different angles of 90o. And that the monoclinic crystals are those that present only an angle other than 90o. By this same principle should the "diclinic" system exist, right? that is, a system formed by crystals with two angles equal to 90° and one different! However, the diclinic crystals are grouped in the triclinic system, and the explanation is very simple: the "diclinic" crystals have the same crystal classes as the triclinic crystals. Therefore, there is no sense in creating a diclinical system, but there is also no sense in using such a confusing nomenclature.

The conventions for nomenclature of crystal systems could then be different, that is, more logical. The names of tetragonal (which have a 4 or -4 symmetry axis), trigonal (which have a 3 or -3 symmetry axis) and hexagonal (which have a 6 or -6 axis) systems appear to have no problems since their name is directly connected to the symmetry elements that characterize them. The other names (cubic, orthorhombic, monoclinic and triclinic) are connected to the unit cells. One option would be to use for these systems the names suggested by Peacock (1950), which relate to the symmetry elements. The cubic system (which has four axes 3 or -3) would now be called tetra-trigonal, the orthorhombic system (which has three 2 or -2 (= m) axes) would be called tri-digonal, the monoclinic (which has one 2 or -2 (= m) axis) would be named digonal system and the triclinic system (which only has 1 or -1(= i) axis) could be renamed monogonal system. These four names are logical and technically appropriate, unlike those that are officially used, but there would certainly be resistance to replacing terms as deeply rooted, as yet occurred in 1950.

29th Jan 2018 21:06 UTCAmir C. Akhavan Expert

I can't say much about the diclinic system and its merits, but apart from that it may appear to some readers as if what you wrote on cell parameters is in conflict with IUC definitions.

However, while it is true that probably all mineralogy textbooks do it wrong and define, for example, triclinic cell parameters as "a ≠ b ≠ c, α ≠ β ≠ γ", the International Tables say that there are "no restrictions" on cell parameters for triclinic crystal system, so a cell with parameters "a=b=c, α = β = γ" is perfectly o.k. for a triclinic mineral, just as you wrote.

Or, to give another example, according to the International Tables, orthorhombic restricts the angles to be 90°, but a, b and c can assume any value, contrary to what most textbooks write.

See, page 15, table 2.1.2.1 "Crystal families, crystal systems, conventional coordinate systems and Bravais lattices in one, two and three dimensions" in International Tables for Crystallography, 2005 Edition.

30th Jan 2018 00:59 UTCJohan Kjellman Expert

Interesting. Is it only deep-rooted habit that has prevented acceptance of this seemingly logic nomenclature change? Apparently crystallographers live with the contradictive character of the nomenclature and obviously they don't obey the faulty definitions. They know that it is the symmetries that have to be obeyed not the axial parameters in order to determine crystal system, and it is evident in the simple symbolism of the lowest symmetry class for each system, 1, 2, 222, 3, 4, 6 and 23. Comparing these with your suggested nomenclature, only the last suggestion seem out of place - would "digonal-trigonal" be the appropriate name? One wonders was there ever any debate after Peacock (1950) regarding the old vs suggested new nomenclature?

cheers

31st Aug 2018 16:20 UTCBenjamin Oelkers

Having done some structure determinations myself, I would like to add that the main point made here is in my opinion not that the used nomenclature is imperfect in a technical sense. Far more important is the realisation stated by Amir above: "Triclinic" does not necessarily mean that all angles and axes are different from one another, but that they can be - or that they are not related by symmetry (which means just the same). This (and the other, usually less severe cases described above) has led to quite a number of wrong symmetry assumptions, most of which will render the data gathered from an XRD experiment "useless" to the less experienced person.

Nevertheless, I personally cannot see that much would be gained by clarifying the names of the crystal systems as proposed. The reason for this is that "monoclinic" and "triclinic" nicely describe the usual behaviour - in most cases, there is no angle of 90° in a triclinic lattice, for example. So while we would gain some definite information on crystal symmetry by adopting the proposed nomenclature, we would lose some useful information "for everyday use".

And as for the cubic system: Just think of "symmetry like a cube or a tetrahedron", and you will be fine. ;-)

Greetings, Benjamin

2nd Sep 2018 19:12 UTCAlfred L. Ostrander

I have spent a number of years studying crystallography. Presently, due to an ongoing study of crystallography with my friend Don Peck, I have fifteen textbooks sitting beside my desk. If I listed the authors, some of you would recognize many of the names as crystallographers of known repute. I can tell you this: each author has tried to simplify the nomenclature and none have been in total agreement. Don groans at this confusion and I just laugh. Austin F. Rogers noted many years ago that over 30 names have been proposed for the (hkl) form in the ditetragonal dipyramidal class 4/mm 2/mm 2/mm. So, I see nothing new proposed here. And I am sure many new proposals will be made in the future.

I agree with those who have noted that the name triclinic means more than many realize. Instead of changing the name the full meaning should be presented, learned and held to. No need for more clutter in the nomenclature. The same goes for other terms. As I have studied crystallography through the tears I have found that many definitions have been short changed by not being presented in their fullness. This is not the fault of the nomenclature or definition but of the author presenting the information. I would prefere to see clear definitions that have been used properly through time rather than creating new terms that may not be any clearer.

I think a part of the problem depends on following a strict adherence to geometric forms followed by the indices or following the recognition of a form to an axial attitude and analogous forms throughout the classes. This can be seen when considering the use of the term prism as a strict form regardless of attitude or the separation of prisms relating to the vertical axis and domes defined as special prisms relating to the lateral axes. Each approach has led to a wide ranging nomenclature. And heaven help the novice student trying to sort out the 1st, 2nd, 3rd and 4th order of the monoclinic system to the 1st, 2nd and 3rd order prisms of the tetragonal, trigonal and hexagonal systems. They just don't mean the same thing. The try to figure out 1st, 2nd and 3rd order pinakoids. And if a pinacoid is really a parallelohedron, why isn't it called that? The list of idiosyncrasies goes on and on.

No matter what, a student interested in the history of crystallography is going to have to learn several of the major systems of nomenclature already in play. You simply cannot read the older authors without doing so. Why should one more system of nomenclature be added?

3rd Sep 2018 01:21 UTCTom Tucker

I loved my crystallography course 50 years ago - but please, no more new names.

5th Sep 2018 03:56 UTCDonald B Peck Expert

I have read this thread. I am not and do not claim to be a crystallographer. However I have studied it both at university and on my own. I have read Peacock's paper and Rogers' paper, and half a dozen texts. Despite a few flaws, to me, Rogers makes the most sense. He uses the nomenclature that is seemingly most common. His conventions are not overly complex (in that I probably do not fully understand them).

I was a teacher (chemistry, physics, and geology), both at the high school and university level. I have developed a specialty (as an amateur) in mineralogy and crystallography, and believe me I am in full sympathy with students trying to learn the details of crystallography. The conventions are maddeningly inconsistent and confusing. I agree with Al Ostrander and Tom Tucker, we do not need any more names for forms or new systems. What we need is to simplify the nomenclature, using short names already commonly used, and develop short, readable, and understandable statements of crystallographic relationships and differences.

Please pardon my rant and my ignorance. I have been struggling with this topic for several months, and am a bit frustrated.

6th Sep 2018 02:33 UTCDoug Daniels

Don - struggling with it for only a few months? I have a piece of paper from1978 saying I know something about geology (and, I got an A in my mineralogy class...), but I still have problems with crystallography. Then again, I haven't spent a lot of time trying to learn it, using the specimens I have on hand. Gotta give a hand to those that have learned about it.

6th Sep 2018 13:53 UTCAlfred L. Ostrander

Doug,

Don and I have been engaged in a rather in depth study comparing different authors approaches as well as group point theory and drafting crystal projections. Of course, nomenclature certainly has come into play. Don has adapted the graphic method of Naumann to computer use and is doing drawings in clinometric projection. Some of this may be "re-inventing" what others have done but we set out to do this and are not backing down. You might say we are looking back at a lot of history. You might also say we have embarked on a graduate level thesis adventure. Don recently commented that if any of his non-mineralogic friends knew where his thoughts were lately they would really consider him "weird". My wife just laughs as this project is keeping me occupied. Take out the trash, mow the yard, study crystallography and take my wife out to dinner. Life is good.

After this, Don will certainly be able to hold his own in any discussion regarding crystallography. Actually, he already can. And we are not done yet...

6th Sep 2018 14:35 UTCKeith Compton 🌟 Manager

Alfred

I love what you and Don are doing

Great articles so far.

As a matter of interest, which Crystalographic book(s) do you recommend (prefer)?

6th Sep 2018 16:14 UTCAlfred L. Ostrander

Keith,

Certaily a work in progress. Don is doing a great job. I'm just the guy bugging Don and most likely baffling him even further with my musings about things of a crystallographic nature.

I have very mixed thoughts on any book to recommend. Each text I have been using follows its own system of nomenclature based on approach. The International System after Fedorov adheres to strict geometric forms. Other systems, like Dana, allow for relationships to the axes for names of forms. Then, everything in between seems to be the order of affairs. I have a large pile, fifteen or more, that are useful. I have several more I might put in the stove this winter.

For a basic introduction, Mineralogy by Sinkankas is good. Any edition, still available as used copies.

For those with a background, I have several:

Mineralogy 5th Edition, Krauss Hunt Ramsdell, 1959. Good for a student or mineralogist. Follows closer to Dana and others rather than the International System. It has very clear definitons of forms. Each class discussion is ended with a well organized chart of forms found in said class. Well illustrated. Roughly 90 pages on crystallogarphy. The rest is standard descriptive mineralogy.

Dana's Textbook of Mineralogy 4th Edition, W. E. Ford. Descriptions of forms are clear. No chart at the end of each class for forms. Many of the drawings are labeled with letters for the forms that are discussed in the text. Roughly 200 pages on crystallography. The rest is standard descriptive mineralogy with lots of crystal drawings.

If you consider yourself advanced, try Crystallography and Practical Crystal Measurement, Volume 1, 1921, A.E.H. Tutton. Great line drawings as in Dana's Textbook and labeled in a similar fashion. Nomenclature for forms is different. Wading through it you can see a mix of several different approaches to nomenclature. Not for the beginner or for anyone easily bored with crystallography.

For now, these are the books I depended on as a student. We used the wooden set of models, 220 plus some, specifically after Dana. I am not sure what I would recommend yet for the International System. I know how this system works but being brought up in the manner of Dana, I find myself treating the international system as a foreign language. That is, I can readily translate it in my head and speak it. Still, it is not my native language. This system seems to be coming into vogue. I will continue to look for a good text.

I am bringing up these older texts as they are reasonably available on ebay and Amazon and are not very expensive at all, including postage. I am as much an historian of crystallography as a crystallographer. Some people may not appreciate these older texts as not in line with modern thinking. I hold to the thinking that if you don't know how you got to where you are how will you know where to go. I will be putting together a thorough list for Don. Stay tuned.

6th Sep 2018 17:04 UTCDonald B Peck Expert

As Al said, we have been working on this project for several months and Al is keeping me out of hot water. It has been quite challenging, and actually, the graphic parts, the clinometric projections have been the the easiest. They take time and are a bit tedious, but Microsoft Publisher has a pretty good graphics capability. It allows placement of lines and plane figures to within one degree of where I want them. It is the multitude of names and conventions that make the task difficult. We are trying to write the articles on the crystal systems so each is independent of the others and for the intermediately experienced collector. I think they get better with each one, and that is making me go back and edit some of the previous ones.

Early on, Erin Devinthal suggested that mindat's rotating 3D models would add imensely to understanding and Jolyon must have agreed, because he did the coding to make it possible almost immediately. Al and I are indebted to them for the idea and for making it possible. If anyone has suggestions, (it is a bit late in the process, but . . .) we are open to suggestions and may be able to edit them into the articles.

Keith, thank you for the comment. Doug, I think we studied mineralogy and crystallography nearly at the same time. My studies were just a few years before yours. My reference "for a few months" relates to the duration of this project. With luck Al and I may finish it by the end of this year. He is a great partner.

The paper that makes the most sense to me is: http://www.minsocam.org/ammin/AM20/AM20_838.pdf A Tabulation of Crystal Forms and Discussion of Form-Names; Auston F. Rogers. A discussion of the Federov Plan, now the International System; a seminal paper. Then follow this with Dana's Textbook of Mineralogy. I also like Mason & Berry's, Elements of Mineralogy. It is a bit incomplete, but a good introduction, especially when followed by Dana's Text.

Don

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